三角和张量积Bézier曲面间相互转换的新方法

New Conversion between Triangular and Tensor-Product Bézier Patches

在计算机辅助几何设计中 ,已有的三角 Bézier曲面和张量积 Bézier曲面间的相互转换算法 ,通常是将一个三角 Bézier曲面转化为三个张量积 Bézier曲面 ,或将一个张量积 Bézier曲面转化为两个三角 Bézier曲面 ,但这样会增加系统存储和显示的负担 .针对这一问题 ,提出了一类新的转换方法 ,即 :将一个三角 Bézier曲面表示为一个张量积 Bézier曲面的 Trimm ed曲面 ,或者将一个张量积 Bézier曲面表示为一个三角 Bézier曲面的 Trimmed曲面 .理论分析和实验结果表明 ,当用基于广义 de Casteljau算法实现转换时 ,新方法与已有方法的数值精度相同 ,而在计算时间和存储量方面只有原来方法的 1/ 3或 1/ 2 .此外 ,新方法有利于在 Open GL的编程环境下显示三角Bézier曲面 .

Triangular and tensor product Bézier patches are widely used in computer aided geometric design. It is important to study the conversion between them for modeling complex shapes, fulfilling data exchanges between CAD systems, etc. In general, a triangular Bézier patch is converted into three tensor product Bézier patches, or a tensor product Bézier patch is converted into two triangular Bézier patches. However, such conversions will increase the burdens of storage and display. In this paper, a new conversion scheme between two types of patches are given, i.e., a triangular Bézier patch is converted into a trimmed surface of tensor product Bézier patch, or a tensor product Bézier patch is converted into a trimmed surface of triangular Bézier patch. Theoretical analysis shows that the runtime and storage of new approach are only 1/2 or 1/3 of that of existing approaches, and experiments show that the numerical accuracy remains unchanged. The additional advantage of new approach is able to display the triangular Bézier patches by using OpenGL API.